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ON THE  COLOURS  OF THICK PLATES.                     163
for R must be the same as if x and y were each equal to zero. We have therefore
[L, Jf] = (ZtoZ3)+c'-c3 + g-i,
and [M, L\ is formed from [L, M] by interchanging the coordinates of L and M. In the above expression e2 has been written for shortness, in place of &2+Z>2. Now supposing c, c,, c2, c3, to be all positive, and denoting by A, B the points in which the first and second surfaces respectively are cut by the -axis of the mirror, we have
which gives, on expanding,
(L to La) = c + ^ - ^ - ^ + M (c, + t) + g-   + M (c, + *)
2- - - 2-c -
We have therefore
Although this formula was obtained on the supposition that the points L, Llt Z2, JS3, AT, Jf1? Jf2, J/<}, lay on the positive side of the plane of xy, it is true independently of that restriction. For when one of the foci Z, Llt L.2> L^, from having been real becomes virtual, or from having been virtual becomes real, the corresponding ordinatc c, GI} c2 H- 1, or c3 changes sign. At the same time, in the expression for the retardation distance passed over is converted into distance saved, and vice versa. Hence in any such expression as (2) the sign of one or more of the lines is changed. But in the expansion of the radical by which the length of such line is expressed, the sign of c, c1? c2 + t, or c3 must be changed at the same time, and therefore no change is required in the expanded expressions.